I've got one who is just turning 11. She, however, loves suspense stories, though she didn't like Harry Potter.

I second the suggestion for the Narnia series. Also the Anne of Green Gables books, and The Girl of the Limberlost. My daughter suggests Sarah, Plain and Tall; The Lemonade Crime; and The Phantom Tollbooth.

How about Witches of Karres by James Schmitz, available from Baen. He wrote that for a girl of about her age, as I recall, and it would make her laugh. (And if you wanted to read it to her, it would make you laugh, as well.)

Enola Holmes series is very good. Skullduggery pleasant and Artemis Fowl are good fantasy books for 11 year old. You would enjoy these too maybe. I think are all at libraries.

I haven't read the Enola Holmes series but while I adore Skulduggery Pleasant and have enjoyed Artemis Fowl, and would otherwise wholeheartedly recommend them for any kid of that age, they're possibly not well suited for someone who has sleep issues and gets anxiety from books with too much excitement (going by Order of the Phoenix being too much). Same goes for other series I'd mention otherwise, like Rick Riordan's Percy Jackson or Kane Chronicles.

I'm trying hard to come up with something that hasn't been mentioned yet, but the majority of books I've read that are aimed at that age group tend to go for "action-packed and exciting".

Oh... how about the Underland Chronicles by Suzanne Collins? I've only read the first one (Gregor the Overlander), admittedly, but found it adventurous while well below the later HP books in the scariness or anxiety-inducing excitement levels. Another possibility might be the Ember series by Jeanne DuPrau, starting with City of Ember.

I haven't read the Enola Holmes series but while I adore Skulduggery Pleasant and have enjoyed Artemis Fowl, and would otherwise wholeheartedly recommend them for any kid of that age, they're possibly not well suited for someone who has sleep issues and gets anxiety from books with too much excitement (going by Order of the Phoenix being too much). Same goes for other series I'd mention otherwise, like Rick Riordan's Percy Jackson or Kane Chronicles.

I'm trying hard to come up with something that hasn't been mentioned yet, but the majority of books I've read that are aimed at that age group tend to go for "action-packed and exciting".

Oh... how about the Underland Chronicles by Suzanne Collins? I've only read the first one (Gregor the Overlander), admittedly, but found it adventurous while well below the later HP books in the scariness or anxiety-inducing excitement levels. Another possibility might be the Ember series by Jeanne DuPrau, starting with City of Ember.

Enola Holmes is Sherlock's younger sister, 12 or 13 I think. Very nice to read and reinforces the idea that children are far more mature and capable than we often see them.,

The Circle of Magic series by Tamora Pierce. Then anything else written by Pierce.
The Enchanted Forest series by Patricia Wrede.

The following might or might not be for readers younger than your daughter, but I remember liking these as a kid (no idea what age I was, though):
E.B. White: Trumpet of the Swan, Stuart Little, etc.
Beverly Cleary: The Mouse and the Motorcycle series, etc.
Ruth Chew: Do-It-Yourself Magic, etc.
Lynne Reid Banks: The Indian in the Cupboard series, etc.

Robin McKinley's Damar duology is great, but she might not want to read it at bedtime. Same for Patricia McKillip's Riddle Master of Hed series. Also consider Wildside by Steven Gould.

Tamora Pierce's books seem to be suitable for that age. I recommend starting with The Song of the Lioness quartet. The first book is Alanna: The First Adventure.

ETA: Oops, looks like BaenSidhe already beat me to it haha. Shame on me for not reading all the responses first. Pierce has 2 universes she writes about. The specific books BaenSidhe and I suggested are of opposite universes.

Abby Carnelia's One and Only Magical Power by David Pogue. (since we are all gadgety here, you probably recognize the name: yes, the NY Times tech writer David Pogue). I read it to my daughter at bedtime, very funny book!